Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models
Marco Di Francesco Donatella Donatelli
In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller-Segel type systems. The approximating systems are either hyperbolic-parabolic or hyperbolic-elliptic. They all feature a nonlinear pressure term arising from a volume filling effect which takes into account the fact that cells do not interpenetrate. The main convergence result relies on energy methods and compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assumptions, we also prove an existence result for initial data which are small perturbation of a constant state. Such result is proven via classical Friedrichs's symmetrization and linearization. In order to simplify the coverage, we restrict to the two-dimensional case with periodical boundary conditions.
keywords: Keller Segel model chemotaxis diffusive relaxation Singular convergence volume filling effect. nonlinear diffusion nonlinear hyperbolic systems
On the diffusive stress relaxation for multidimensional viscoelasticity
Donatella Donatelli Corrado Lattanzio
This paper deals with the rigorous study of the diffusive stress relaxation in the multidimensional system arising in the mathematical modeling of viscoelastic materials. The control of an appropriate high order energy shall lead to the proof of that limit in Sobolev space. It is shown also as the same result can be obtained in terms of relative modulate energies.
keywords: viscoelasticity Diffusive relaxation limits
On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions
Donatella Donatelli Eduard Feireisl Antonín Novotný
We study the low Mach number limit for the compressible Navier-Stokes system supplemented with Navier's boundary condition on an unbounded domain with compact boundary. Our main result asserts that the velocities converge pointwise to a solenoidal vector field - a weak solution of the incompressible Navier-Stokes system - while the fluid density becomes constant. The proof is based on a variant of local energy decay property for the underlying acoustic equation established by Kato.
keywords: singular limits compressible fluids low Mach number unbounded domains. Navier-Stokes equations

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